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2 5.3 Multiplying Polynomials: Special Products

3 What You Will Learn  Find products with monomial multipliers  Multiplying binomials using the Distributive Property and the FOIL Method  Multiply polynomials using a horizontal or vertical format  Identify and use special binomial products

4 Monomial Multipliers

5 To multiply polynomials, you use many of the rules for simplifying algebraic expressions. 1. The Distributive Property 2. Combining like terms 3. Removing symbols of grouping 4. Rules of exponents The simplest type of polynomial multiplication involves a monomial multiplier. The product is obtained by direct application of the Distributive Property.

6 Monomial Multipliers For instance, to multiply the monomial x by the polynomial (2x + 5), multiply each term of the polynomial by x. (x)(2x + 5) = (x)(2x) + (x)(5) = 2x 2 + 5x

7 Example 1 – Finding Products with Monomial Multipliers Find each product. a. (3x – 7)(–2x) b. 3x 2 (5x – x 3 + 2) c. (–x)(2x 2 – 3x) Solution: a. (3x – 7)(–2x) = 3x(–2x) – 7(–2x) = –6x x Distributive Property Write in standard form.

8 cont’d b. 3x 2 (5x – x 3 + 2) = (3x 2 )(5x) – (3x 2 )(x 3 ) + (3x 2 )(2) = 15x 3 – 3x 5 + 6x 2 = –3x x 3 + 6x 2 c. (–x)(2x 2 – 3x) = (–x)(2x 2 ) – (–x)(3x) = –2x 3 + 3x 2 Distributive Property Rules of exponents Write in standard form. Distributive Property Write in standard form. Example 1 – Finding Products with Monomial Multipliers

9 Multiplying Binomials

10 Multiplying Binomials To multiply two binomials, you can use both (left and right) forms of the Distributive Property. For example, if you treat the binomial (5x + 7) as a single quantity, you can multiply (3x – 2) by (5x + 7) as follows. (3x – 2)(5x + 7) = 3x(5x + 7) – 2(5x + 7) = (3x)(5x) + (3x)(7) – (2)(5x) – 2(7) = 15x x – 10x – 14 = 15x x – 14

11 Multiplying Binomials With practice, you should be able to multiply two binomials without writing out all of the steps above. In fact, the four products in the boxes above suggest that you can write the product of two binomials in just one step. This is called the FOIL Method. Note that the words first, outer, inner, and last refer to the positions of the terms in the original product.

12 Example 2 – Multiplying Binomials with the Distributive Property Use the Distributive Property to find each product. a. (x – 1)(x + 5) b. (2x + 3) (x – 2) Solution: a. (x – 1)(x + 5) = x(x + 5) – 1(x + 5) = x 2 + 5x – x – 5 = x 2 + (5x – x) – 5 = x 2 + 4x – 5 Right Distributive Property Left Distributive Property Group like terms. Combine like terms.

13 cont’d Example 2 – Multiplying Binomials with the Distributive Property b. (2x + 3)(x – 2) = 2x(x – 2) + 3(x – 2) = 2x 2 – 4x + 3x – 6 = 2x 2 + (–4x + 3x) – 6 = 2x 2 – x – 6 Right Distributive Property Left Distributive Property Group like terms. Combine like terms.

14 Example 3 – Multiplying Binomials using the FOIL Method Use the FOIL Method to find each product. a. (x + 4)(x – 4) b. (3x + 5)(2x + 1) Solution: F O I L a. (x + 4)(x – 4) = x 2 – 4x + 4x – 16 = x 2 – 16 Note that the outer and inner products add up to zero. Combine like terms.

15 Example 3 – Multiplying Binomials using the FOIL Method cont’d F O I L b. (3x + 5)(2x + 1) = 6x 2 + 3x + 10x + 5 = 6x x + 5 Combine like terms.

16 Example 4 – A Geometric Model of a Polynomial Product Use the geometric model to show that x 2 + 3x + 2 = (x + 1)(x + 2)

17 Example 4 – A Geometric Model of a Polynomial Product cont’d Solution The left part of the model shows that the sum of the areas of the six rectangle is x 2 + (x + x + x) + (1 + 1) = x 2 + 3x + 2 The right part of the model shows that the area of the rectangle is (x + 1)(x + 2) = x 2 + 2x + x + 2 = x 2 + 3x + 2 So, x 2 + 3x + 2 = (x + 1)(x + 2)

18 Example 5 – Simplifying a Polynomial Expression Simplify the expression and write the result in standard form (4x + 5) 2 Solution (4x + 5) 2 = (4x + 5)(4x + 5) Repeated multiplication = 16x x + 20x + 25 Use FOIL Method = 16x x + 25 Combine like terms

19 Example 6 – Simplifying a Polynomial Expression Simplify the expression and write the result in standard form (3x 2 – 2)(4x + 7) – (4x) 2 Solution (3x 2 – 2)(4x + 7) – (4x) 2 = 12x x 2 – 8x – 14 – (4x) 2 Use FOIL Method = 12x x 2 – 8x – 14 – 16x 2 Square monomial = 12x 3 + 5x 2 – 8x – 14 Combine like terms

20 Multiplying Polynomials

21 Multiplying Polynomials The FOIL Method for multiplying two binomials is simply a device for guaranteeing that each term of one binomial is multiplied by each term of the other binomial. (ax + b)(cx + d) = ax(cx) + ax(d) + b(cx) + b(d) F O I L This same rule applies to the product of any two polynomials: each term of one polynomial must be multiplied by each term of the other polynomial. This can be accomplished using either a horizontal or a vertical format.

22 Example 7 – Multiplying Polynomials Horizontally Use a horizontal format to find each product. a. (x – 4)(x 2 – 4x + 2) b. (2x 2 – 7x + 1)(4x + 3) Solution: a. (x – 4)(x 2 – 4x + 2) = x(x 2 – 4x + 2) – 4(x 2 – 4x + 2) = x 3 – 4x 2 + 2x – 4x x – 8 = x 3 – 8x x – 8 Combine like terms. Distributive Property

23 Example 7 – Multiplying Polynomials Horizontally cont’d b. (2x 2 – 7x + 1)(4x + 3) = (2x 2 – 7x + 1)(4x) + (2x 2 – 7x + 1)(3) = 8x 3 – 28x 2 + 4x + 6x 2 – 21x + 3 = 8x 3 – 22x 2 – 17x + 3 Combine like terms. Distributive Property

24 Example 10 – Raising a Polynomial to a Power Use two steps to expand (x – 3) 3 Solution: Step 1: (x – 3) 2 = (x – 3)(x – 3) = x 2 – 3x – 3x + 9 = x 2 – 6x + 9 Step 2: (x 2 – 6x + 9)(x – 3)= (x 2 – 6x + 9)(x) – (x 2 – 6x + 9)(3) = x 3 – 6x 2 + 9x – 3x x – 27 = x 3 – 9x x – 27 So, (x – 3) 3 = x 3 – 9x x – 27 Combine like terms Use FOIL Method Repeated multiplication

25 Special Products

26 Special Products Some binomial products, such as those in Example 3(a), has special forms that occur frequently in algebra. The product (x + 4)(x – 4) is called a product of the sum and difference of two terms. With such products, the two middle terms cancel, as follows. (x + 4)(x – 4) = x 2 – 4x + 4x – 16 = x 2 – 16 Sum and difference of two terms Product has no middle term.

27 Special Products Another common type of product is the square of a binomial. (4x + 5) 2 = (4x + 5)(4x + 5) = 16x x + 20x + 25 = 16x x + 25 Square of a binomial Use FOIL Method. Middle term is twice the product of the terms of the binomial.

28 Special Products In general, when a binomial is squared, the resulting middle term is always twice the product of the two terms. (a + b) 2 = a 2 + 2(ab) + b 2 Be sure to include the middle term. For instance, (a + b) 2 is not equal to a 2 + b 2.

29 Special Products

30 Example 11 – Finding Special Products a.(5x – 6)(5x + 6) = (5x) 2 – (6) 2 = 25x 2 – 36 b.(3x + 7) 2 = (3x) 2 + 2(3x)(7) + (7) 2 = 9x x + 14 c.(4x + 9) 2 = (4x) 2 + 2(4x)(9) + (9) 2 = 16x x + 81 d.(6 + 5x 2 ) 2 = (4) 2 – 2(6)(5x 2 ) + (5x 2 ) 2 = 36 – 60x 2 + (5) 2 (x 2 ) 2 = 36 – 60x x 4

31 Example 12 – Finding the Dimensions of a Golf Tee A landscaper wants to reshape a square tee area for the ninth hole of a golf course. The new tee area will have one side 2 feet longer and the adjacent side 6 feet longer than the original tee. The area of the new tee will be 204 square feet greater than the area of the original tee. What are the dimensions of the original tee?

32 Solution Verbal Model: Labels: Original length = original width = x (feet) Original area = x2 (square feet) New length = x + 6 (feet) New width = x + 2 (feet) Equation:(x + 6)(x + 2) = x Write equation x 2 + 8x + 12 = x Multiply factors 8x + 12 = 204 Subtract x 2 from each side 8x = 192 Subtract 12 from each side x = 24 Divide each side by 8 cont’d Example 12 – Finding the Dimensions of a Golf Tee